# Properties

 Label 189.2.g.b Level $189$ Weight $2$ Character orbit 189.g Analytic conductor $1.509$ Analytic rank $0$ Dimension $10$ CM no Inner twists $2$

# Related objects

Show commands: Magma / PariGP / SageMath

## Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [189,2,Mod(100,189)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(189, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([4, 2]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("189.100");

S:= CuspForms(chi, 2);

N := Newforms(S);

 Level: $$N$$ $$=$$ $$189 = 3^{3} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 189.g (of order $$3$$, degree $$2$$, not minimal)

## Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

 Self dual: no Analytic conductor: $$1.50917259820$$ Analytic rank: $$0$$ Dimension: $$10$$ Relative dimension: $$5$$ over $$\Q(\zeta_{3})$$ Coefficient field: 10.0.991381711347.1 comment: defining polynomial  gp: f.mod \\ as an extension of the character field Defining polynomial: $$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$$ x^10 - 2*x^9 + 9*x^8 - 8*x^7 + 40*x^6 - 36*x^5 + 90*x^4 - 3*x^3 + 36*x^2 - 9*x + 9 Coefficient ring: $$\Z[a_1, \ldots, a_{4}]$$ Coefficient ring index: $$3^{2}$$ Twist minimal: no (minimal twist has level 63) Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

## $q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\ldots,\beta_{9}$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q - \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{4} + (\beta_{9} - \beta_{2} + 1) q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - \beta_{8} - \beta_{4} + 1) q^{8}+O(q^{10})$$ q - b1 * q^2 + (b7 + b6 - b3) * q^4 + (b9 - b2 + 1) * q^5 + (-b8 + b6 - b4 - b1 + 1) * q^7 + (-b8 - b4 + 1) * q^8 $$q - \beta_1 q^{2} + (\beta_{7} + \beta_{6} - \beta_{3}) q^{4} + (\beta_{9} - \beta_{2} + 1) q^{5} + ( - \beta_{8} + \beta_{6} - \beta_{4} + \cdots + 1) q^{7}+ \cdots + ( - 2 \beta_{9} + 3 \beta_{8} + \cdots - 10) q^{98}+O(q^{100})$$ q - b1 * q^2 + (b7 + b6 - b3) * q^4 + (b9 - b2 + 1) * q^5 + (-b8 + b6 - b4 - b1 + 1) * q^7 + (-b8 - b4 + 1) * q^8 + (-2*b6 + b4 + b2 - 2) * q^10 + (-b8 - b4 - b3 + 1) * q^11 + (-b6 - b2 - b1 - 1) * q^13 + (b6 - b5 + b4 - b3 + b2 + b1 - 2) * q^14 + (b7 + b4 + b2) * q^16 + (-b7 - 3*b6 + b1 - 3) * q^17 + (b9 + b8 - b7 - b6 + 2*b5 + b3) * q^19 + (2*b8 + b7 + 2*b6 - b3) * q^20 + (-b7 - b6 + b2 + b1 - 1) * q^22 + (-b9 + 2*b8 + 2*b4 + 2*b3 + b2) * q^23 + (2*b9 - b5 - b3 - 2*b2 + b1) * q^25 + (b9 - b8 + b7 + b6 + 2*b5 - b3) * q^26 + (-2*b9 + 2*b6 - b5 - b4 - b3 + 2*b1) * q^28 + (b8 - 2*b7 + b6 + b5 + 2*b3) * q^29 + (-b9 + b8 - b7 + b6 + b3) * q^31 + (-2*b9 - b8 + b7 + b6 - 3*b5 - b3) * q^32 + (b8 - b7 - 2*b6 + 4*b5 + b3) * q^34 + (-b9 - b8 - b6 - b5 + 2*b3 + b2 + b1 - 1) * q^35 + (-2*b8 - 2*b5) * q^37 + (b8 + 2*b5 + b4 + 3*b3 - 2*b1 + 5) * q^38 + (-b8 - b5 - b4 + 2*b3 + b1 + 1) * q^40 + (-b7 - b4 - 2*b2 - b1) * q^41 + (-3*b9 + b8 - b7 + 3*b6 - 2*b5 + b3) * q^43 + (-b9 + b7 - 2*b6 + b5 - b3) * q^44 + (2*b7 + 4*b6 - b4 - 3*b2 - 5*b1 + 4) * q^46 + (2*b7 - 4*b6 - b4 - b2 - b1 - 4) * q^47 + (b9 - b7 - 3*b5 - b3 + b2 + 3*b1 + 1) * q^49 + (-b7 - 6*b6 + b4 + 2*b2 + 3*b1 - 6) * q^50 + (b8 - 2*b5 + b4 + b3 + 2*b1 + 1) * q^52 + (5*b6 - 2*b2 - b1 + 5) * q^53 + (b9 + b8 - b5 + b4 + b3 - b2 + b1) * q^55 + (b9 - b8 - b7 - b6 - b5 - b4 + b1 + 4) * q^56 + (b9 + b8 + 2*b5 + b4 + 2*b3 - b2 - 2*b1 + 3) * q^58 + (b9 + b8 + b7 + 6*b6 + b5 - b3) * q^59 + (b7 - 2*b6 - b4 + b2 - b1 - 2) * q^61 + (2*b9 - b8 + 2*b5 - b4 + b3 - 2*b2 - 2*b1 + 3) * q^62 + (-b9 - b5 - 2*b3 + b2 + b1 - 6) * q^64 + (-b7 + b6 + b4 + b2 + b1 + 1) * q^65 + (-b9 - b8 + 2*b7 - b6 + 4*b5 - 2*b3) * q^67 + (b9 + 2*b5 + 3*b3 - b2 - 2*b1 + 7) * q^68 + (-b9 + b8 - b7 - 3*b6 + 2*b4 - b3 - 2*b1 - 5) * q^70 + (-2*b9 + 3*b8 - 2*b5 + 3*b4 - b3 + 2*b2 + 2*b1 - 2) * q^71 + (-b7 + 4*b6 - 3*b4 + 4) * q^73 + (-2*b9 + 2*b8 - 2*b5 + 2*b4 - 4*b3 + 2*b2 + 2*b1 - 10) * q^74 + (b7 + 3*b6 + b2 - 5*b1 + 3) * q^76 + (-b9 - b8 - 2*b7 - 2*b4 + 2*b2 + b1 + 3) * q^77 + (-2*b7 - 3*b6 + b2 + 4*b1 - 3) * q^79 + (-3*b6 - b4 + b2 - 2*b1 - 3) * q^80 + (3*b9 - 2*b8 - 2*b6 + 4*b5) * q^82 + (-2*b9 - 4*b8 + 2*b7 + b6 - 2*b3) * q^83 + (-b7 - 2*b6 + b4 + b2 - 2) * q^85 + (4*b9 - 3*b8 + 2*b5 - 3*b4 - b3 - 4*b2 - 2*b1 + 1) * q^86 + (-b9 - 2*b8 - 2*b4 - b3 + b2 + 4) * q^88 + (b9 + 2*b8 + 7*b6 - 2*b5) * q^89 + (-b8 + 2*b7 - 2*b6 + b5 + 2*b4 - b3 + 2*b2 + 2*b1 - 2) * q^91 + (2*b9 - 2*b8 + 5*b6 - 2*b5) * q^92 + (2*b9 - 4*b8 - 3*b6 + 4*b5) * q^94 + (b8 + 2*b7 - 2*b6 - 2*b3) * q^95 + (2*b9 - b8 + 4*b7 + 2*b6 + b5 - 4*b3) * q^97 + (-2*b9 + 3*b8 - 3*b7 - 5*b6 - b5 + b2 + b1 - 10) * q^98 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$10 q - 2 q^{2} - 4 q^{4} + 8 q^{5} - q^{7} + 6 q^{8}+O(q^{10})$$ 10 * q - 2 * q^2 - 4 * q^4 + 8 * q^5 - q^7 + 6 * q^8 $$10 q - 2 q^{2} - 4 q^{4} + 8 q^{5} - q^{7} + 6 q^{8} - 7 q^{10} + 8 q^{11} - 8 q^{13} - 16 q^{14} + 2 q^{16} - 12 q^{17} + q^{19} - 5 q^{20} - q^{22} + 6 q^{23} + 2 q^{25} - 11 q^{26} - 2 q^{28} - 7 q^{29} - 3 q^{31} + 2 q^{32} + 3 q^{34} - 5 q^{35} + 40 q^{38} + 6 q^{40} - 5 q^{41} - 7 q^{43} + 10 q^{44} + 3 q^{46} - 27 q^{47} + 25 q^{49} - 19 q^{50} + 20 q^{52} + 21 q^{53} + 4 q^{55} + 45 q^{56} + 20 q^{58} - 30 q^{59} - 14 q^{61} + 12 q^{62} - 50 q^{64} + 11 q^{65} - 2 q^{67} + 54 q^{68} - 29 q^{70} + 6 q^{71} + 15 q^{73} - 72 q^{74} + 5 q^{76} + 31 q^{77} - 4 q^{79} - 20 q^{80} - 5 q^{82} - 9 q^{83} - 6 q^{85} - 16 q^{86} + 36 q^{88} - 28 q^{89} - 4 q^{91} - 27 q^{92} - 3 q^{94} + 14 q^{95} - 12 q^{97} - 59 q^{98}+O(q^{100})$$ 10 * q - 2 * q^2 - 4 * q^4 + 8 * q^5 - q^7 + 6 * q^8 - 7 * q^10 + 8 * q^11 - 8 * q^13 - 16 * q^14 + 2 * q^16 - 12 * q^17 + q^19 - 5 * q^20 - q^22 + 6 * q^23 + 2 * q^25 - 11 * q^26 - 2 * q^28 - 7 * q^29 - 3 * q^31 + 2 * q^32 + 3 * q^34 - 5 * q^35 + 40 * q^38 + 6 * q^40 - 5 * q^41 - 7 * q^43 + 10 * q^44 + 3 * q^46 - 27 * q^47 + 25 * q^49 - 19 * q^50 + 20 * q^52 + 21 * q^53 + 4 * q^55 + 45 * q^56 + 20 * q^58 - 30 * q^59 - 14 * q^61 + 12 * q^62 - 50 * q^64 + 11 * q^65 - 2 * q^67 + 54 * q^68 - 29 * q^70 + 6 * q^71 + 15 * q^73 - 72 * q^74 + 5 * q^76 + 31 * q^77 - 4 * q^79 - 20 * q^80 - 5 * q^82 - 9 * q^83 - 6 * q^85 - 16 * q^86 + 36 * q^88 - 28 * q^89 - 4 * q^91 - 27 * q^92 - 3 * q^94 + 14 * q^95 - 12 * q^97 - 59 * q^98

Basis of coefficient ring in terms of a root $$\nu$$ of $$x^{10} - 2x^{9} + 9x^{8} - 8x^{7} + 40x^{6} - 36x^{5} + 90x^{4} - 3x^{3} + 36x^{2} - 9x + 9$$ :

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$( -\nu^{9} - 9\nu^{8} - 3\nu^{7} - 61\nu^{6} - 72\nu^{5} - 282\nu^{4} - 204\nu^{3} - 387\nu^{2} - 873\nu - 117 ) / 189$$ (-v^9 - 9*v^8 - 3*v^7 - 61*v^6 - 72*v^5 - 282*v^4 - 204*v^3 - 387*v^2 - 873*v - 117) / 189 $$\beta_{3}$$ $$=$$ $$( 7\nu^{9} - 12\nu^{8} + 48\nu^{7} - 23\nu^{6} + 204\nu^{5} - 240\nu^{4} + 303\nu^{3} - 108\nu^{2} + 36\nu - 1557 ) / 567$$ (7*v^9 - 12*v^8 + 48*v^7 - 23*v^6 + 204*v^5 - 240*v^4 + 303*v^3 - 108*v^2 + 36*v - 1557) / 567 $$\beta_{4}$$ $$=$$ $$( 2\nu^{9} - \nu^{8} + 12\nu^{7} + 8\nu^{6} + 68\nu^{5} + 30\nu^{4} + 123\nu^{3} + 204\nu^{2} + 270\nu + 63 ) / 63$$ (2*v^9 - v^8 + 12*v^7 + 8*v^6 + 68*v^5 + 30*v^4 + 123*v^3 + 204*v^2 + 270*v + 63) / 63 $$\beta_{5}$$ $$=$$ $$( 16 \nu^{9} - 39 \nu^{8} + 156 \nu^{7} - 176 \nu^{6} + 663 \nu^{5} - 780 \nu^{4} + 1680 \nu^{3} + \cdots - 180 ) / 567$$ (16*v^9 - 39*v^8 + 156*v^7 - 176*v^6 + 663*v^5 - 780*v^4 + 1680*v^3 - 351*v^2 + 684*v - 180) / 567 $$\beta_{6}$$ $$=$$ $$( 20 \nu^{9} - 24 \nu^{8} + 141 \nu^{7} - 4 \nu^{6} + 624 \nu^{5} - 57 \nu^{4} + 1020 \nu^{3} + \cdots - 63 ) / 567$$ (20*v^9 - 24*v^8 + 141*v^7 - 4*v^6 + 624*v^5 - 57*v^4 + 1020*v^3 + 1620*v^2 + 369*v - 63) / 567 $$\beta_{7}$$ $$=$$ $$( - 53 \nu^{9} + 60 \nu^{8} - 375 \nu^{7} - 11 \nu^{6} - 1668 \nu^{5} - 69 \nu^{4} - 2757 \nu^{3} + \cdots - 1368 ) / 567$$ (-53*v^9 + 60*v^8 - 375*v^7 - 11*v^6 - 1668*v^5 - 69*v^4 - 2757*v^3 - 4401*v^2 - 1071*v - 1368) / 567 $$\beta_{8}$$ $$=$$ $$( - 82 \nu^{9} + 165 \nu^{8} - 732 \nu^{7} + 632 \nu^{6} - 3264 \nu^{5} + 2850 \nu^{4} - 7260 \nu^{3} + \cdots + 720 ) / 567$$ (-82*v^9 + 165*v^8 - 732*v^7 + 632*v^6 - 3264*v^5 + 2850*v^4 - 7260*v^3 - 432*v^2 - 2898*v + 720) / 567 $$\beta_{9}$$ $$=$$ $$( - 91 \nu^{9} + 174 \nu^{8} - 813 \nu^{7} + 704 \nu^{6} - 3633 \nu^{5} + 3174 \nu^{4} - 8070 \nu^{3} + \cdots + 801 ) / 567$$ (-91*v^9 + 174*v^8 - 813*v^7 + 704*v^6 - 3633*v^5 + 3174*v^4 - 8070*v^3 + 648*v^2 - 3222*v + 801) / 567
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{7} + 3\beta_{6} - \beta_{3}$$ b7 + 3*b6 - b3 $$\nu^{3}$$ $$=$$ $$\beta_{8} + 4\beta_{5} + \beta_{4} - 4\beta _1 - 1$$ b8 + 4*b5 + b4 - 4*b1 - 1 $$\nu^{4}$$ $$=$$ $$-5\beta_{7} - 14\beta_{6} + \beta_{4} + \beta_{2} - 14$$ -5*b7 - 14*b6 + b4 + b2 - 14 $$\nu^{5}$$ $$=$$ $$2\beta_{9} - 7\beta_{8} - \beta_{7} - 9\beta_{6} - 17\beta_{5} + \beta_{3}$$ 2*b9 - 7*b8 - b7 - 9*b6 - 17*b5 + b3 $$\nu^{6}$$ $$=$$ $$9\beta_{9} - 10\beta_{8} - \beta_{5} - 10\beta_{4} + 24\beta_{3} - 9\beta_{2} + \beta _1 + 70$$ 9*b9 - 10*b8 - b5 - 10*b4 + 24*b3 - 9*b2 + b1 + 70 $$\nu^{7}$$ $$=$$ $$11\beta_{7} + 65\beta_{6} - 43\beta_{4} - 19\beta_{2} + 75\beta _1 + 65$$ 11*b7 + 65*b6 - 43*b4 - 19*b2 + 75*b1 + 65 $$\nu^{8}$$ $$=$$ $$-62\beta_{9} + 73\beta_{8} + 118\beta_{7} + 360\beta_{6} + 14\beta_{5} - 118\beta_{3}$$ -62*b9 + 73*b8 + 118*b7 + 360*b6 + 14*b5 - 118*b3 $$\nu^{9}$$ $$=$$ $$-135\beta_{9} + 253\beta_{8} + 343\beta_{5} + 253\beta_{4} - 87\beta_{3} + 135\beta_{2} - 343\beta _1 - 430$$ -135*b9 + 253*b8 + 343*b5 + 253*b4 - 87*b3 + 135*b2 - 343*b1 - 430

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/189\mathbb{Z}\right)^\times$$.

 $$n$$ $$29$$ $$136$$ $$\chi(n)$$ $$-1 - \beta_{6}$$ $$\beta_{6}$$

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field

gp: mfembed(f)

Label   $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
100.1
 1.19343 + 2.06709i 0.920620 + 1.59456i 0.247934 + 0.429435i −0.335166 − 0.580525i −1.02682 − 1.77851i 1.19343 − 2.06709i 0.920620 − 1.59456i 0.247934 − 0.429435i −0.335166 + 0.580525i −1.02682 + 1.77851i
−1.19343 2.06709i 0 −1.84857 + 3.20182i 2.92087 0 2.35742 1.20106i 4.05086 0 −3.48586 6.03769i
100.2 −0.920620 1.59456i 0 −0.695084 + 1.20392i −1.33475 0 −2.54347 0.728536i −1.12285 0 1.22880 + 2.12835i
100.3 −0.247934 0.429435i 0 0.877057 1.51911i 3.69258 0 −2.60948 + 0.436591i −1.86155 0 −0.915516 1.58572i
100.4 0.335166 + 0.580525i 0 0.775327 1.34291i −1.42494 0 2.21529 + 1.44655i 2.38012 0 −0.477591 0.827212i
100.5 1.02682 + 1.77851i 0 −1.10873 + 1.92038i 0.146246 0 0.0802402 + 2.64453i −0.446582 0 0.150168 + 0.260099i
172.1 −1.19343 + 2.06709i 0 −1.84857 3.20182i 2.92087 0 2.35742 + 1.20106i 4.05086 0 −3.48586 + 6.03769i
172.2 −0.920620 + 1.59456i 0 −0.695084 1.20392i −1.33475 0 −2.54347 + 0.728536i −1.12285 0 1.22880 2.12835i
172.3 −0.247934 + 0.429435i 0 0.877057 + 1.51911i 3.69258 0 −2.60948 0.436591i −1.86155 0 −0.915516 + 1.58572i
172.4 0.335166 0.580525i 0 0.775327 + 1.34291i −1.42494 0 2.21529 1.44655i 2.38012 0 −0.477591 + 0.827212i
172.5 1.02682 1.77851i 0 −1.10873 1.92038i 0.146246 0 0.0802402 2.64453i −0.446582 0 0.150168 0.260099i
 $$n$$: e.g. 2-40 or 990-1000 Embeddings: e.g. 1-3 or 100.5 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
63.g even 3 1 inner

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 189.2.g.b 10
3.b odd 2 1 63.2.g.b 10
4.b odd 2 1 3024.2.t.i 10
7.b odd 2 1 1323.2.g.f 10
7.c even 3 1 189.2.h.b 10
7.c even 3 1 1323.2.f.e 10
7.d odd 6 1 1323.2.f.f 10
7.d odd 6 1 1323.2.h.f 10
9.c even 3 1 189.2.h.b 10
9.c even 3 1 567.2.e.e 10
9.d odd 6 1 63.2.h.b yes 10
9.d odd 6 1 567.2.e.f 10
12.b even 2 1 1008.2.t.i 10
21.c even 2 1 441.2.g.f 10
21.g even 6 1 441.2.f.f 10
21.g even 6 1 441.2.h.f 10
21.h odd 6 1 63.2.h.b yes 10
21.h odd 6 1 441.2.f.e 10
28.g odd 6 1 3024.2.q.i 10
36.f odd 6 1 3024.2.q.i 10
36.h even 6 1 1008.2.q.i 10
63.g even 3 1 inner 189.2.g.b 10
63.g even 3 1 3969.2.a.bc 5
63.h even 3 1 567.2.e.e 10
63.h even 3 1 1323.2.f.e 10
63.i even 6 1 441.2.f.f 10
63.j odd 6 1 441.2.f.e 10
63.j odd 6 1 567.2.e.f 10
63.k odd 6 1 1323.2.g.f 10
63.k odd 6 1 3969.2.a.bb 5
63.l odd 6 1 1323.2.h.f 10
63.n odd 6 1 63.2.g.b 10
63.n odd 6 1 3969.2.a.z 5
63.o even 6 1 441.2.h.f 10
63.s even 6 1 441.2.g.f 10
63.s even 6 1 3969.2.a.ba 5
63.t odd 6 1 1323.2.f.f 10
84.n even 6 1 1008.2.q.i 10
252.o even 6 1 1008.2.t.i 10
252.bl odd 6 1 3024.2.t.i 10

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
63.2.g.b 10 3.b odd 2 1
63.2.g.b 10 63.n odd 6 1
63.2.h.b yes 10 9.d odd 6 1
63.2.h.b yes 10 21.h odd 6 1
189.2.g.b 10 1.a even 1 1 trivial
189.2.g.b 10 63.g even 3 1 inner
189.2.h.b 10 7.c even 3 1
189.2.h.b 10 9.c even 3 1
441.2.f.e 10 21.h odd 6 1
441.2.f.e 10 63.j odd 6 1
441.2.f.f 10 21.g even 6 1
441.2.f.f 10 63.i even 6 1
441.2.g.f 10 21.c even 2 1
441.2.g.f 10 63.s even 6 1
441.2.h.f 10 21.g even 6 1
441.2.h.f 10 63.o even 6 1
567.2.e.e 10 9.c even 3 1
567.2.e.e 10 63.h even 3 1
567.2.e.f 10 9.d odd 6 1
567.2.e.f 10 63.j odd 6 1
1008.2.q.i 10 36.h even 6 1
1008.2.q.i 10 84.n even 6 1
1008.2.t.i 10 12.b even 2 1
1008.2.t.i 10 252.o even 6 1
1323.2.f.e 10 7.c even 3 1
1323.2.f.e 10 63.h even 3 1
1323.2.f.f 10 7.d odd 6 1
1323.2.f.f 10 63.t odd 6 1
1323.2.g.f 10 7.b odd 2 1
1323.2.g.f 10 63.k odd 6 1
1323.2.h.f 10 7.d odd 6 1
1323.2.h.f 10 63.l odd 6 1
3024.2.q.i 10 28.g odd 6 1
3024.2.q.i 10 36.f odd 6 1
3024.2.t.i 10 4.b odd 2 1
3024.2.t.i 10 252.bl odd 6 1
3969.2.a.z 5 63.n odd 6 1
3969.2.a.ba 5 63.s even 6 1
3969.2.a.bb 5 63.k odd 6 1
3969.2.a.bc 5 63.g even 3 1

## Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator $$T_{2}^{10} + 2T_{2}^{9} + 9T_{2}^{8} + 8T_{2}^{7} + 40T_{2}^{6} + 36T_{2}^{5} + 90T_{2}^{4} + 3T_{2}^{3} + 36T_{2}^{2} + 9T_{2} + 9$$ acting on $$S_{2}^{\mathrm{new}}(189, [\chi])$$.

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{10} + 2 T^{9} + \cdots + 9$$
$3$ $$T^{10}$$
$5$ $$(T^{5} - 4 T^{4} - 5 T^{3} + \cdots - 3)^{2}$$
$7$ $$T^{10} + T^{9} + \cdots + 16807$$
$11$ $$(T^{5} - 4 T^{4} - 8 T^{3} + \cdots - 15)^{2}$$
$13$ $$T^{10} + 8 T^{9} + \cdots + 25$$
$17$ $$T^{10} + 12 T^{9} + \cdots + 81$$
$19$ $$T^{10} - T^{9} + \cdots + 185761$$
$23$ $$(T^{5} - 3 T^{4} + \cdots + 1611)^{2}$$
$29$ $$T^{10} + 7 T^{9} + \cdots + 81$$
$31$ $$T^{10} + 3 T^{9} + \cdots + 81225$$
$37$ $$T^{10} + 96 T^{8} + \cdots + 82944$$
$41$ $$T^{10} + 5 T^{9} + \cdots + 2025$$
$43$ $$T^{10} + 7 T^{9} + \cdots + 687241$$
$47$ $$T^{10} + 27 T^{9} + \cdots + 43758225$$
$53$ $$T^{10} - 21 T^{9} + \cdots + 178929$$
$59$ $$T^{10} + 30 T^{9} + \cdots + 31640625$$
$61$ $$T^{10} + 14 T^{9} + \cdots + 1$$
$67$ $$T^{10} + 2 T^{9} + \cdots + 50708641$$
$71$ $$(T^{5} - 3 T^{4} - 168 T^{3} + \cdots + 81)^{2}$$
$73$ $$T^{10} - 15 T^{9} + \cdots + 772641$$
$79$ $$T^{10} + 4 T^{9} + \cdots + 37249$$
$83$ $$T^{10} + \cdots + 218123361$$
$89$ $$T^{10} + 28 T^{9} + \cdots + 7080921$$
$97$ $$T^{10} + \cdots + 2307745521$$